Optimal. Leaf size=49 \[ \frac {2 \sqrt {a x^2+b x^3}}{3 b}-\frac {4 a \sqrt {a x^2+b x^3}}{3 b^2 x} \]
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Rubi [A] time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2016, 1588} \begin {gather*} \frac {2 \sqrt {a x^2+b x^3}}{3 b}-\frac {4 a \sqrt {a x^2+b x^3}}{3 b^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 1588
Rule 2016
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {a x^2+b x^3}} \, dx &=\frac {2 \sqrt {a x^2+b x^3}}{3 b}-\frac {(2 a) \int \frac {x}{\sqrt {a x^2+b x^3}} \, dx}{3 b}\\ &=\frac {2 \sqrt {a x^2+b x^3}}{3 b}-\frac {4 a \sqrt {a x^2+b x^3}}{3 b^2 x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 0.61 \begin {gather*} \frac {2 (b x-2 a) \sqrt {x^2 (a+b x)}}{3 b^2 x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 32, normalized size = 0.65 \begin {gather*} \frac {2 (b x-2 a) \sqrt {a x^2+b x^3}}{3 b^2 x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 28, normalized size = 0.57 \begin {gather*} \frac {2 \, \sqrt {b x^{3} + a x^{2}} {\left (b x - 2 \, a\right )}}{3 \, b^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {b x^{3} + a x^{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 33, normalized size = 0.67 \begin {gather*} -\frac {2 \left (b x +a \right ) \left (-b x +2 a \right ) x}{3 \sqrt {b \,x^{3}+a \,x^{2}}\, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 30, normalized size = 0.61 \begin {gather*} \frac {2 \, {\left (b^{2} x^{2} - a b x - 2 \, a^{2}\right )}}{3 \, \sqrt {b x + a} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.16, size = 31, normalized size = 0.63 \begin {gather*} -\frac {\left (\frac {4\,a}{3\,b^2}-\frac {2\,x}{3\,b}\right )\,\sqrt {b\,x^3+a\,x^2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {x^{2} \left (a + b x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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